Parallelization (mathematics)
In mathematics, a parallelization[1] of a manifold [math]\displaystyle{ M\, }[/math] of dimension n is a set of n global smooth linearly independent vector fields.
Formal definition
Given a manifold [math]\displaystyle{ M\, }[/math] of dimension n, a parallelization of [math]\displaystyle{ M\, }[/math] is a set [math]\displaystyle{ \{X_1, \dots,X_n\} }[/math] of n smooth vector fields defined on all of [math]\displaystyle{ M\, }[/math] such that for every [math]\displaystyle{ p\in M\, }[/math] the set [math]\displaystyle{ \{X_1(p), \dots,X_n(p)\} }[/math] is a basis of [math]\displaystyle{ T_pM\, }[/math], where [math]\displaystyle{ T_pM\, }[/math] denotes the fiber over [math]\displaystyle{ p\, }[/math] of the tangent vector bundle [math]\displaystyle{ TM\, }[/math].
A manifold is called parallelizable whenever it admits a parallelization.
Examples
- Every Lie group is a parallelizable manifold.
- The product of parallelizable manifolds is parallelizable.
- Every affine space, considered as manifold, is parallelizable.
Properties
Proposition. A manifold [math]\displaystyle{ M\, }[/math] is parallelizable iff there is a diffeomorphism [math]\displaystyle{ \phi \colon TM \longrightarrow M\times {\mathbb R^n}\, }[/math] such that the first projection of [math]\displaystyle{ \phi\, }[/math] is [math]\displaystyle{ \tau_{M}\colon TM \longrightarrow M\, }[/math] and for each [math]\displaystyle{ p\in M\, }[/math] the second factor—restricted to [math]\displaystyle{ T_pM\, }[/math]—is a linear map [math]\displaystyle{ \phi_{p} \colon T_pM \rightarrow {\mathbb R^n}\, }[/math].
In other words, [math]\displaystyle{ M\, }[/math] is parallelizable if and only if [math]\displaystyle{ \tau_{M}\colon TM \longrightarrow M\, }[/math] is a trivial bundle. For example, suppose that [math]\displaystyle{ M\, }[/math] is an open subset of [math]\displaystyle{ {\mathbb R^n}\, }[/math], i.e., an open submanifold of [math]\displaystyle{ {\mathbb R^n}\, }[/math]. Then [math]\displaystyle{ TM\, }[/math] is equal to [math]\displaystyle{ M\times {\mathbb R^n}\, }[/math], and [math]\displaystyle{ M\, }[/math] is clearly parallelizable.[2]
See also
- Chart (topology)
- Differentiable manifold
- Frame bundle
- Orthonormal frame bundle
- Principal bundle
- Connection (mathematics)
- G-structure
- Web (differential geometry)
Notes
References
- Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6, https://archive.org/details/tensoranalysison00bish
- Milnor, J.W.; Stasheff, J.D. (1974), Characteristic Classes, Princeton University Press
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